Multimorphological Crystallization of Shish-Kebab Structures in

Jul 16, 2014 - ABSTRACT: A model is presented for the crystallization kinetics of flow-induced shish-kebab structures in isotactic polypropylene. The ...
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Multimorphological Crystallization of Shish-Kebab Structures in Isotactic Polypropylene: Quantitative Modeling of Parent−Daughter Crystallization Kinetics Peter C. Roozemond,†,‡ Zhe Ma,† Kunpeng Cui,§ Liangbin Li,§ and Gerrit W. M. Peters*,† †

Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands National Synchrotron Radiation Lab and College of Nuclear Science and Technology, CAS Key Laboratory of Soft Matter Chemistry, University of Science and Technology of China, Hefei, China

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ABSTRACT: A model is presented for the crystallization kinetics of flow-induced shish-kebab structures in isotactic polypropylene. The model accounts for two phenomena that affect the crystal growth of kebab structures. First, the temperature of the material increases due to latent heat release. Second, polymer chains are deformed in such a way that growth of kebabs (with their c-axis in flow direction) is promoted. Furthermore, we incorporate crystallization kinetics of the daughter morphology which nucleate on the (010) lateral surface of kebabs. The model is validated with in-situ wide-angle X-ray diffraction data from an in-house-developed extensional flow device and a modified multipass rheometer used to apply a flow pulse in a slit flow geometry. Excellent agreement is found between the model and experimental data, in terms of crystallization kinetics as well as parent/daughter ratio. The well-known Avrami model, in which it is assumed that crystal growth rate is constant, is widely used to analyze (quasi-)isothermal flowinduced crystallization experiments. We show that, because the growth rate of kebabs is strongly time dependent due to temperature and orientation effects, this analysis fails for experiments like the ones presented in this study. This is manifested in noninteger Avrami coefficients typically smaller than 2, which have been observed in a number of different studies. The current model explains these observations.

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device,6 sandwich-type shear cell,12,13 or Linkam-type or Couette shear cell.1,2,8 Well-defined experiments with high deformation rates, relevant to processing (∼1000 s−1), can be accomplished most straightforwardly in geometries without free surfaces, e.g., a slit flow, although this geometry has the inherent disadvantage of a nonhomogeneous shear rate distribution. The famous shish-kebab structure, consisting of an extendedchain crystal core (shish) on which disklike folded chain crystals (kebabs) nucleate, has been documented in the seminal works by Keller and Pennings.14−17 In iPP, the subject of the present study, an additional morphology, known as daughters, grows epitaxially on the (010) surface of kebabs (in this case also known as parents).4,5,7 Although qualitatively very well researched, the crystallization kinetics after strong flows are relatively poorly understood. The typical way of modeling the growth of kebabs, as cylinders growing in radial direction, is using the Schneider rate equations4,18,19 combined with the Kolmogorov−Avrami equation.20,21 However, this approach fails nearly always,4,6−8 which is typically exhibited in Avrami plots slowing slopes of noninteger values, smaller than two.

INTRODUCTION Semicrystalline polymers, such as isotactic polypropylene (iPP) and polyethylene (PE), are widely used in a large variety of products. Because these polymers are processed from the molten state, the flow field acts as an unavoidable factor that can change the kinetics and morphology of crystallization and consequently affect the ultimate structure and properties. In order to optimize processing conditions, as well as create products with tailored properties, it is vital to have good understanding of the mechanism behind flow-induced nucleation and crystal growth. A wealth of experimental work has been performed on the subject of flow-induced crystallization.1−8 Most of the experiments were done using the wellknown short-term flow protocol9 in order to obtain welldefined isothermal experimental conditions in which structure formation is separated from crystal growth. Consequently, the effect of flow on structure formation can be studied in detail because the crystallization kinetics in quiescent conditions can be characterized thoroughly in separate experiments (cf. ref 10). Crystallization kinetics are usually probed in situ with birefringence3,9 or either small-angle X-ray scattering (SAXS), wide-angle X-ray diffraction (WAXD), or a combination of the two.1,2,4−7,11 These types of experiments can be performed in various geometries, for example a slit flow,4,5,7,9 extensional flow © 2014 American Chemical Society

Received: May 28, 2014 Revised: June 23, 2014 Published: July 16, 2014 5152

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Table 1. Experimental Conditions in the MPRa

The stronger the flow, typically, the lower the Avrami coefficient. Some authors ascribe this observation to a two-step process (shish growing longitudinally and kebabs that only start growing after a certain induction time).22 Other authors hypothesized an altered crystal growth rate during flow with respect to quiescent conditions (cf. the Supporting Information of ref 4). In this paper we present a model that accurately captures noninteger Avrami coefficients as well as parent/daughter crystallization kinetics. It is important to note there that we do not model the structure formation during flow. Instead, the structure generated by flow is an initial condition from which our model calculates crystallization kinetics. To this end we have adapted the Schneider rate equations to account for temperature increase due to latent heat release as well as an increased growth rate of parent crystals during and shortly after flow. The model is validated with experimental data from extension flow as well as slit flow, combined with in-situ WAXD. These experiments are presented in the Experimental Section. In the Model section we present the governing equations of the model, followed by validation in the Results section. Finally, in the Conclusions section we briefly reiterate the most important conclusions from this work.

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shear rate at wall γ̇w [s−1]

shear stress at wall σw [×105 Pa]

60 80 100 120

0.25 0.25 0.25 0.23

370 500 635 769

1.21 1.35 1.47 1.57

Wall shear rate and wall shear stress were calculated at steady state using a Carreau−Yasuda model for the shear rate dependence of the viscosity. where i can be parent or daughter. A∞,i depends on the detector and the thickness of the sample (extension) or shear layer (MPR). Because this thickness is unknown (for the extension experiments one can easily make an estimation, but this is not as straightforward as one might think; see the Appendix), we determined A∞,p, the 110 peak area for 100% space filling of parent crystals, separately for each flow condition. A∞,d is related to A∞,p via a geometrical correction factor, which depends on scattering angle.5,25 In this work we do not perform the full geometrical correction. Instead, we take a constant factor to relate the two, A∞,d = cgeometryA∞,d. The areas of parent and daughter crystals are now given by A p ,110(t ) = ξp(t )A∞ , p

Ad ,110(t ) = ξd(t )cgeometryA∞ , d

(2)

We determined one factor for each type of experiment. For the extension experiments we found cgeometry = 3.7 and for the MPR experiments cgeometry = 3.3. These values are close to the range cgeometry = 3.7−4.4 found by Ma et al.26 All experiments with the MPR were performed twice: once with a Frelon detector with an exposure time of 2 s and azimuthal range of >90° and once with a Pilatus detector that has an exposure time of 0.03 s and azimuthal range of 90°. Patterns for background correction were acquired for each detector at corresponding acquisition times using an empty load geometry. In addition, a dark current (no X-ray exposure) was subtracted for the Frelon recorded patterns to correct for readout noise. For the data collected with the Pilatus detector, the area of the parent reflection is given by the area underneath the (isotropic) baseline subtracted scattering pattern. Regarding the patterns acquired with the Frelon detector and the data obtained with the extensional flow device, azimuthal scans of the (110) reflection were fitted by Lorentzian peaks to calculate the area of both the parent and daughter lamellae reflections. Examples of procedures for both detectors are given in Figure 1. For further information about the WAXD setups the reader is referred to refs 6, 11, and 23. Depth Sectioning. The inherent disadvantage of the slit flow geometry is the nonuniformity of shear stress and rate along the thickness. Consequently, there will also be a variation in crystalline structure in thickness direction; highly oriented, densely packed shish near the wall and isotropic structure near the center of the slit. Therefore, the measured cystallinity from WAXD is effectively an average over the slit thickness. Fernandez-Ballester et al. proposed the “depth-sectioning” method5 to correct for this. We use the method to extract the crystallization kinetics of only the shear layer near the wall from the diffraction patterns, which otherwise probe an average of the crystallization kinetics over the slit thickness. The concept of the depth-sectioning method departs from the linear variation of local shear stress with position from the center x. Take two experiments, with piston speeds vpist,1 and vpist,2 and corresponding wall shear stresses σw,1 and σw,2, with σw,2 > σw,1. Then the stress history for experiment 1 for positions 0 < x < d/2 will be the same as the stress history in experiment 2 for positions 0 < x < (σw,1/σw,2)(d/2). If crystal structure is a function of only shear stress, the contribution to the

In order to validate our model, we compare calculations with data from experiments that were already presented elsewhere.6,11 We use wide-angle X-ray diffraction (WAXD) data measured in these experiments that were not yet published. We discuss experiments obtained in two different settings. In both cases the setup consists of a flow device combined with insitu WAXD to measure crystallinity. In one case the flow setup is a uniaxial extensional flow device.23 The other is piston driven slit flow.11 In both cases the material was a commercial iPP. For the extensional flow experiments the material was kindly supplied by SABIC-Europe, with a weightaverage molecular weight Mw = 720 kg/mol and a number-average molecular weight Mn = 150 kg/mol. For the slit flow experiments the material is Borealis HD601CF with Mw = 365 kg/mol and Mn = 68 kg/mol. The short-term flow protocol9 was used; first, thermomechanical history is erased above the melting point, after which the material is cooled down to the experimental temperature. At the experimental temperature, a flow is applied with a duration short compared to the time needed to complete crystallization, and crystallization during and after flow was monitored under isothermal conditions. Extensional Flow. The extensional flow experiments were performed at the National Synchrotron Radiation Lab in Hefei, China, using an in-house developed extensional rheometer.23 The temperature was 138 °C in all cases. The data in this study were obtained at three strain rates; 3.1, 12.6, and 25.1 s−1. For each strain rate three flow durations were investigated, to give a total strain of 2.0, 2.5, and 3.0 for all strain rates. Slit Flow. The slit flow experiments11 were performed at the Dutch-Belgian beamline BM26B (DUBBLE) at the European Synchrotron Radiation Facility ESRF) in Grenoble, France. The flow cell was operated on a modified multipass rheometer (MPR),24 which in this case was solely used to subject the material to a flow pulse in a slit. The experimental temperature was 145 °C in all cases. More information on the experimental conditions is given in Table 1. WAXD Analysis. In this paper we compare crystallinity of the parent and daughter species from our calculations and experiments. The crystallinity from experiments is obtained by dividing the area below the 110 peak of the parents and daughters by the area at complete space filling Ai (t ) A∞ , i

flow duration [s]

a

EXPERIMENTAL SECTION

ξi(t ) =

piston speed [mm/s]

(1) 5153

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Figure 1. Azimuthal scans of the (110) diffraction of WAXD patterns (inserted) obtained with (a) Pilatus detector and (b) Frelon detector. Dashed black curves indicate the fitted Lorentzian functions. Flow direction is vertical.

Figure 2. Area below 110 peak for parent crystals for different piston speeds: (a) before depth sectioning; (b) after depth sectioning. WAXD signal from the crystalline structure in experiment 2 for (σw,1/σw,2)(d/2) < x < d/2 is given by σw ,1 A 2−1 = A 2 − A1 σw ,2 (3) with A1(t) and A2(t) the 110 areas from WAXD for experiments 1 and 2, respectively. To make a fair comparison between different flow conditions, one has to normalize the signal for layer thickness. The final 110 areas from depth sectioning then become A2 − A 2−1 =

σw ,1 A σw ,2 1

1 − σw ,1/σw ,2

(4)

In this way we extract crystallization kinetics from only highly oriented shish-kebab from the data, which would otherwise also contain the crystallization kinetics of the core layer, which will also contain crystallization of spherulites. As is shown in Table 1, the ratios of shear stresses between different flow conditions is approximately 0.95 in all cases. Hence, the layers of which we calculate the crystallization kinetics typically fill 5% of the thickness of the slit. In this paper we have used data from four piston speeds for validation purposes. The areas below the 110 peak of parent crystals are given in Figure 2a. Corresponding steady-state shear rates and shear stresses at the wall, calculated using the finite element code presented in ref 27, are given in Table 1. Using the depth-sectioning technique, we have converted these data into information about three layers. This is shown in Figure 2b. Note: the depth sectioning method is just an approximation in this case, as (1) pressure effects are not taken into account, (2) we

Figure 3. Evolution of pressure drop over the slit for different piston speeds. neglect start-up behavior, which shows different time scales for different piston speeds, and (3) the shear times for 120 and 100 mm/s differ slightly (0.25 and 0.23 s, respectively). However, as can be see in Figure 3, which shows the pressure drop over the slit geometry, the time scale for start-up of flow is close for all piston speeds, as is the pressure. Moreover, Figure 2b shows that all data end more or less in a plateau, indicating that indeed the crystallization kinetics of morphologies with less dense crystal structure have been filtered out. 5154

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Figure 4. Schematic depiction of a sample in (a) the extensional flow device and (b) the MPR.

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perature distribution was calculated with a finite element code which is described in detail in ref 27. In the extension experiments the deformation rates are much smaller, and thus this term can be neglected. Boundary Conditions. The heat balance requires two boundary conditions on the temperature. In the center of the sample (at x = 0), no flux boundary conditions are prescribed:

MODEL Paramount to accurately describing crystallization kinetics is accurately calculating the crystal growth, which changes in time due to (1) orientation of the material and (2) temperature, the latter of which increases during crystallization due to latent heat release. In this section we present the modeling framework to do so. Geometry. The equations presented in this work are solved on a 1D geometry representing half of the sample, as depicted in Figure 4. The X-ray beam traverses across the sample in the x-direction. The center of the sample is at x = 0. At x = d/2 the polymer is in contact with the surroundings: in the case of extension experiments the polymer−air interface and in the slit flow experiments the polymer−steel interface. At both these points we must prescribe appropriate boundary conditions on the temperature. The thickness of the sample in the slit flow geometry is always the same, d/2 = 0.75 mm, whereas it starts at d/2 = 0.5 mm and decreases with increasing strain in the extension experiments. As was shown by Nielsen et al.,28 the sample dimensions govern if the deformation is uniaxial or biaxial extension and hence how the thickness decreases exactly with strain. We come back to this in the Appendix. Energy Equation. To calculate the temperature distribution in the material the heat balance is solved ρcp

∂T ∂ 2T ∂ξ = λ 2 + ρχ∞ ΔH + σ :D ∂t ∂t ∂x

∂T =0 ∂x

For the extension experiments, a Robin boundary condition is prescribed at the polymer−air interface (x = d/2) λ

T = T∞

Table 2. Material Parameters for the Heat Balance symbol

value [unit]

ρp cp,p λp χ∞ ΔH ρs cp,s λs

800 [kg/m3] 3157 [J/(kg K)] 0.11 [W/mK] 65 [%] 207 [J/g] 8000 [kg/m3] 670 [J/(kg K)] 80 [W/mK]

(7)

(8)

Crystallization Kinetics. Strong flows create fibrillar crystalline structures (shish). The length per unit volume of shish, denoted by L, is assumed to be constant after cessation of flow. So-called “kebabs”, also known as parent lamellae, grow radially outward from the shish. On their (010) lateral surface another species of lamellae, known as daughters, nucleate.5,30 Figure 5 depicts the three morphologies in a schematic way.

with ρ, cp, λ, χ∞, and ΔH material parameters as given in Table 2. The latent heat of crystallization ΔH is multiplied with the

quantity

∂T = h(T − T∞) ∂x

where T∞ is the temperature of the air in the oven and h = 100 W/(m2 K) is the typical heat transfer coefficient of convected air.29 The MPR has thermocouples embedded in the steel, 0.5 mm away from the polymer (see Figure 4b). The temperature of these thermocouples remained constant during flow and subsequent crystallization. We use this information for the boundary condition in the slit flow geometry. The energy balance is also solved for the steel between the polymer and the thermocouples, and we prescribe a Dirichlet boundary condition at the place of the thermocouples:

(5)

density (polymer) heat capacity (polymer) heat conduction coefficient (polymer) final crystallinity heat of crystallization density (steel) heat capacity (steel) heat conduction coefficient (steel)

(6)

final crystallinity of the material χ∞ to account for the fact that the material is not fully crystalline. T is temperature, t is time, and ξ is crystalline volume fraction, the modeling of which is described in the Crystallization Kinetics section. Heating due to viscous dissipation is accounted for by the term σ:D. Because we calculate crystallization kinetics after flow, this term is zero during crystallization. However, viscous dissipation does affect the initial temperature distribution in the slit flow experiments. Therefore, the initial tem-

Figure 5. Schematic depiction of a shish (gray) with parent kebabs (red) and daughter lamellae (blue). The surface area of kebab on which both parent and daughter morphology can grow is denoted by ψ1,p. 5155

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Crystallization Kinetics. The Schneider rate equations19 have been well established to accurately describe crystallization kinetics in quiescent conditions. Similar to the approach taken by Eder,18 we have taken the Schneider rate equations as a point of departure to formulate a model that calculates the volume fractions of parents as well as daughter crystals. In this paper we propose an addition to these equations for the daughter species. For the parents we have

Crystal Growth Rate. Because of the speed at which crystallization progresses in the conditions of interest for this paper, the heat released during crystallization cannot be diffused fast enough to keep the sample at isothermal conditions. Therefore, the temperature dependence the crystal growth rate needs to be accounted for. The growth rate in quiescent conditions can be calculated with10 Gq = Gref exp( −cG(T − Tref )2 )

(9)

ψ2, p = 4πL tot

where Gq is the crystal growth rate in quiescent conditions, and Gref, cG, and Tref are parameters given in Table 3. These parameters were

∂tψ1, p = Gpψ2, p

Table 3. Material Parameters for the Crystallization Kinetics10

∂tψ0, p = Gpf p ψ1, p

quantity

symbol

value [unit]

maximum crystal growth rate growth rate temperature dependence reference temperature

Gref cG Tref

4.5 [μm/s] 2.3 × 10−3 [1/K] 363 [K]

Here Ltot denotes the total length of shish per unit volume, and Gp is the crystal growth rate of parent species. ψ2,p, ψ1,p, and ψ0,p are measures for the shish length per unit volume, the surface area of kebabs, and undisturbed volume of kebabs, respectively. The volume fraction of the shish itself is neglected because it is very small compared to the total volume of shish kebabs (the radius of a shish is in the order of 10 nm33 while the radius of a shish kebab grows up to about 100 nm; cf. refs 27 and 34). From our picture of the shish−parent−daughter morphology (Figure 5), it follows that daughter lamellae nucleate on the surface of the parents. Therefore, we propose that the evolution of the volume fraction of daughters scales with the area of the parents

measured for the grade of iPP that was used in the MPR experiments. Since the crystal growth rate of iPP homopolymer shows little dependence on molecular weight,10,31,32 we also use these parameters for the calculations on the extension experiments. Additionally, some authors suggest the growth of parent lamellae is promoted during the flow pulse and relaxation afterward, because chains are oriented in the c-axis direction of their unit cells.4,22 In time, this effect would relax and crystal growth rate would go to the quiescent value. Furthermore, one could imagine that chains that are tethered to the shish backbone, so-called “hairs”,33 crystallize at higher rate than chains in the melt because their mobility is restricted. This would result in a growth rate that depends on kebab radius instead of time. At the moment we cannot distinguish which of the two effects (or both) plays a role. Therefore, we account for the combination of the two with an empirical relation. During flow the growth rate of parents is increased with a factor μflow. At the start of crystallization, coinciding with the end of flow, this effect relaxes due to chains relaxing toward their equilibrium conformation. This happens with a time scale λG Gp(t , T ) = Gq(T )[1 + μflow exp(−t /λG)]

∂tψ0, d = Gdfd ψ1, p

(13)

where ψ0,d is the undisturbed volume fraction of daughters. The total surface area of kebabs, denoted by ψ1,p (see Figure 5), is obtained from eq 12. In our model, the lateral surface of the parents acts as nucleation site for both parents and daughters. Here too it is important that during flow the crystallization of parent crystals is promoted. The parent and daughter morphologies compete for the same nucleation sites. To account for the favorability of one or the other, these nucleation sites are allocated to either parent or daughter morphology based on their their momentary growth rates:

(10)

f p = f p 0 (T )

where Gp is the growth rate of parent lamellae, μflow is the additional growth rate due to flow, λG is the relevant time scale, and t is the time after the cessation of flow. In the present experiments, we found μflow = 4 (in the same order as ref 4, which hypothesized Δμflow = 10). The time scale with which the effect relaxes λG is 6 s in the extension experiments and 9 s in the slit flow experiments, which are in the same order as the relaxation time for an average mode or the relaxation time for chain stretch of the high-molecular-weight tail. The time scale for relaxation of this effect λG is higher in the slit flow experiments, which is unexpected because the material in these experiments has a lower molecular weight than the material in the extension experiments. At the moment we cannot offer an explanation for this observation. The flow-induced conformation makes crystallization in the daughter morphology less preferable. However, because growth of daughter crystals only becomes noticeable when the chains have relaxed to their equilibrium conformations, we take the growth rate for this species always equal to the quiescent growth rate Gd(T ) = Gq(T )

(12)

fd = fd 0 (T )

Gp Gp + Gd

Gd Gp + Gd

(14)

where f p0 and fd0 determine the parent/daughter ratio in quiescent conditions, which may depend on temperature.35,36 In experiments with conditions relevant to the current experiments, it has been observed that the parent/daughter ratio is close to unity for low flow rates.5 Therefore, we take f p0 = fd0 = 1 in all cases. Because f p and fd depend on growth rate, during flow, when the crystal growth rate of parents is increased with respect to the quiescent value, more surface of the kebabs is assigned to nucleate parent crystals. A similar model for the allocation of nucleation sites to different crystal phases of iPP was also used by van Drongelen et al.10 It is important to remark that the proposed mechanism is a minimal model to test the influence of changing growth rate and daughters nucleating on parents; it may be an oversimplification of the actual physics. For example, the time dependence of the growth rate of course does not need to be a simple decaying exponential function and might also depend on

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Figure 6. Crystallinity for parents (a) and daughters (b) in time from simulations with varying shish length. Insets show Avrami plots.

Figure 7. (a) Parent/daughter ratio for different shish lengths per unit volume. (b) Temperature evolution in the center of the sample.

the current model are (1) crystallization kinetics that do not adhere to standard Avrami kinetics and (2) the ratio of parent to daughter crystals which increases with flow strength. The first of these is caused by the fact that crystal growth rate is not constant; it changes due to orientation or stretch of chains and temperature, which increases due to latent heat release. The second phenomenon is a result of the fact that parent and daughters compete for the same volume, combined with the effect of flow on the growth rate of parents. The above-mentioned features are illustrated by the results presented in this section. The conditions and geometry are those of the extension experiments described in the Geometry section. The shish length per unit volume Ltot is varied. Figure 6 shows the evolution of crystallinity of parents (a) and daughters (b). The insets show Avrami plots. Figure 7a shows the parent/ daughter ratio, i.e., ξp(t)/ξd(t). Figure 7b shows the temperature in the center point of the sample. The trends described above are clearly visible; with increasing shish density, crystal growth progresses faster. Therefore, a larger part of the space is filled during or shortly after flow, when the growth rate of parent crystals is higher than the growth rate of daughter crystals. Consequently, the final parent/daughter ratio increases. Moreover, due to the changing growth rate of parents and increasing temperature, the Avrami coefficients are smaller than two. For the lowest shish density shown, a negligible part of space is filled during this time. As a result, the parent/daughter ratio is equal to unity. Furthermore, the crystallization progresses slowly enough that the sample can

position as the chains near the shish backbone are farther from their random equilibrium conformation. In terms of morphology, the model does not account for distortion from cylindrical symmetry, e.g., by daughter lamellae protruding from kebabs. However, as the long spacing (limiting the distance that daughter lamellae can protrude) is in the order of 10 nm,27 and the final radius is in the order of 100 nm,34 we expect the latter effect to be minor. To correct for impingement, we use the well-known Kolmogorov−Avrami equation20 ξ = 1 − exp( −ψ0, p − ψ0, d)

(15)

where ξ is the crystalline volume fraction or space filling. From eq 15 follows that the respective volume fractions of parents and daughters are given by ∂tξp = (1 − ξ)∂tψ0, p

∂tξd = (1 − ξ)∂tψ0, d

(16)

The calculated parent/daughter ratio is now given by ξp/ξd.

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RESULTS In this section we present results from calculations with the current model. First, we show some simulations to highlight the important physics in the model. Next, we compare crystallinity and parent/daughter ratio from our model to experimental data. Influence of Input Parameters on Modeling Results. The key experimental observations that can be explained with 5157

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Figure 8. Area below 110 peak for parents (top row) and daughters (bottom row). Figures from left to right show increasing strain rate. Different symbols/line types are different strains. Symbols show measurements, and lines indicate calculations.

Figure 9. Parent/daughter ratio for experiments with different strain rates. Symbols show measurements, and lines indicate calculations.

earlier, this is a result of the increased growth rate of parents during flow (eq 10); during a short time after flow the growth of parents is faster than growth of daughters. If the majority of the space is filled during this time, parents will be more abundant than daughters because they compete for the same volume. Figure 10 shows the shish length per unit volume for all experiments determined in this way. The values shown are in the order of magnitude of what is expected for such flow conditions and increase with increasing strain and strain rate, as expected.34,37 Slit Flow. In this section experimental data obtained in the MPR are compared with calculations with the model. The shear rates in these experiments are exceedingly high (see Table 1). Consequently, viscous heating has to be taken into account because it significantly affects the temperature of the material at the start of crystallization. We calculated this temperature

lose nearly all the latent heat released during crystallization to the surrounding air, resulting in an Avrami coefficient of close to two. Validation. In this section we compare calculations and experiments. It is important to stress that, once μflow and λG have been determined, there is one adjustable parameter in the model for each flow condition: the shish length per unit volume Ltot. Therefore, this model can be used not only to explain crystallization kinetics but also to extract information about shish density from the experimental data. Extension Experiments. Calculations and experiments are compared in terms of area of the 110 peak in Figure 8. The calculations capture the experimental data quite accurately. Note that for higher strain rates and strains the final volume fraction of parents increases at the cost of the daughters. As we pointed out 5158

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After depth sectioning, we have data for the crystallization experiments in the shear layer(s) for three experiments. We take the shish length per unit volume to be constant within each shear layer. The shish length within each equivalent shear layer was the same across all piston speeds. Outside the shear layers no crystallization takes place. The shish length is fit to each experiment separately, starting with the lowest piston speed. For higher piston speeds, there are multiple shear layers (see Figure 11b). This is important because the heat release in the shear layers close to the center slow down the crystallization kinetics the outermost shear layer. Note that we fit the shish length to the crystallization kinetics of the parents. The crystallization kinetics of the daughters and parent/daughter ratio are predictions. The crystallization kinetics for both parents and daughters from experiments and calculations are compared in Figure 12. Figure 13a shows the parent/daughter ratio. The crystallization kinetics as well as the evolution of the parent/daughter ratio are quantitatively captured by the model. Figure 13b depicts the shish length for each flow condition. The values show the trend that we expect, i.e., increasing with piston speed. These values,

Figure 10. Values of the shish length per unit volume Ltot for different strains and strain rates, obtained from fitting the calculations to experimental data. Lines are to guide the eye.

distribution with the finite element code described in ref 27 and used this as the initial value of the temperature in the slit. The initial temperature distribution is shown in Figure 11a.

Figure 11. (a) Temperature distribution due to viscous dissipation in the slit after flow, calculated with a finite element code. (b) Schematic depiction of shear layers. Piston speed increases from top to bottom.

Figure 12. A110 in the MPR experiments for (a) parents and (b) daughters. Symbols show measurements, and lines show calculations. Insets are Avrami plots. 5159

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Figure 13. (a) Parent/daughter ratio for the MPR experiments. Symbols show measurements, and lines show calculations. (b) Specific shish length for MPR calculations.

Figure 14. Final peak areas of the parent crystals Ap,∞ in the extension experiments. (a) shows the uncorrected values, (b) shows the values corrected for a decrease in thickness from uniaxial deformation, and (c) shows the values corrected for a decrease in thickness from biaxial stretching. Lines are to guide the eye.

Ltot = 1012−1013 m/m3, correspond to shish that are in the order of 300 nm to 1 μm apart. Seki et al. found roughly the same values.34 At long times in the weakest flow condition there is a large discrepancy between the crystallization kinetics of the daughters in calculations and experiments; the experimental data do not show a plateau whereas the calculations do. Possibly the cause of this can be found in crystallization outside the shear layer which was not completely filtered out by depth sectioning.

Consequently, the crystal growth rate changes during crystallization. In this case the classical Avrami analysis no longer applies, resulting in the widely observed noninteger Avrami coefficients. Second, parent and daughter lamellae both nucleate on the surface area of shish kebabs, and they compete for the same volume. During and shortly after flow, crystallization of parent crystals is promoted. Although this effect relaxes quite rapidly after flow (time scale in the order of 10 s), the space filled during this time contains considerably more parents than daughters. Therefore, the parent/daughter ratio increases with shish density and therefore with flow strength/duration.

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CONCLUSIONS We presented a model that, for the first time, accurately captures multimorphological crystallization kinetics of shishkebab structures in iPP. The model provides explanations for two features widely observed in experiments. First, the temperature changes because of latent heat release, and flow deforms chains in such a way that crystallization of parent morphology (with c-axis in flow direction) is promoted.

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APPENDIX. FINAL PEAK AREAS IN EXTENSION The area below the 110 peak of the parent crystals at 100% space filling, A∞,p, for the extension experiments deserves some further investigation. These are shown for all flow conditions in Figure 14a. It is commonly observed5,11 that A∞,p increases with 5160

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increasing strain. However, our data show it decreases with increasing strain. The cause of this observation can be found in the decreasing thickness of the sample with increasing strain. Hence, one would expect that a clear trend emerges when these are corrected for the decreasing thickness with increasing strain, for uniaxial deformation given by d = d0e−ε /2

(17)

The areas after correction are shown in Figure 14b. Even after this correction, no clear trends become visible. However, the thickness calculated using eq 17 might not be the actual thickness of the sample. The group of Hassager performed some tests with a similar extensional rheometer as the present one and found that already for aspect ratios (height/thickness) in the order of 10, the deformation is closer to biaxial rather than uniaxial.28 As the aspect ratio in the current experiments is 18, this is something that has to be taken into account. Therefore, we also tried normalizing the final areas with a thickness that would result from biaxial stretching: d = d 0 e −ε

(18)

These results are shown in Figure 14c. Now we do observe a clear trend: the peak of parents increases with increasing strain. This indicates that indeed the deformation is not perfectly uniaxial and probably at higher strains is closer to biaxial. Therefore, the thickness of the sample, which is used as dimension of the grid to solve the energy balance, might be inaccurate. This does have an effect, but calculations with sample thickness decreasing according to biaxial deformation have shown that this effect becomes noticeable only at the highest strains, and it can be corrected for by slightly increasing L.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (G.W.M.P.). Present Address ‡

DSM Ahead, PO Box 18, 6160 MD Geleen, The Netherlands

Notes

The authors declare no competing financial interest.

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